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Results for “Basic Reproduction Number”

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PubMed2004

Rational behavioral response and the transmission of STDs.

The susceptible-infected (SI) model is extended by allowing for individual optimal choices of self-protective actions against infection, where agents differ with respect to preferences and costs of self-protection. It is shown that a unique endemic equilibrium prevalence exists when the basic reproductive number of a STD is strictly greater than unity, and that the disease-free equilibrium is the unique steady state equilibrium when the basic reproductive number is less than or equal to one. Unlike in models that take individual behavior as given and fixed, the endemic equilibrium prevalence need not vary monotonically with respect to the basic reproductive number. Specifically, with endogenously determined self-protective behavior, a reduction in the basic reproductive number may in fact increase the endemic equilibrium prevalence. The global stability of the endemic steady state is established for the case of a homogeneous population by showing that, for any non-zero initial disease prevalence, there exists an equilibrium path which converges to the endemic steady state.

Humans↗

PubMed2003

Competitive exclusion and coexistence for pathogens in an epidemic model with variable population size.

We study an SIR epidemic model with a variable host population size. We prove that if the model parameters satisfy certain inequalities then competition between n pathogens for a single host leads to exclusion of all pathogens except the one with the largest basic reproduction number. It is shown that a knowledge of the basic reproduction numbers is necessary but not sufficient for determining competitive exclusion. Numerical results illustrate that these inequalities are sufficient but not necessary for competitive exclusion to occur. In addition, an example is given which shows that if such inequalities are not satisfied then coexistence may occur.

Algorithms↗

PubMed1995

The effect of household distribution on transmission and control of highly infectious diseases.

Two epidemic threshold parameters are derived for the spread of a highly infectious disease in a community of households, where a household is any group whose members have frequent contacts with each other. It is assumed that the infection of any member of a household results in the infection of all susceptible members of that household. The threshold parameters have simple expressions in terms of the mean household size and the mean and variance of the number of susceptibles per household. They provide a basic reproduction number R0 for the spread of infection from individual to individual and a basic reproduction number RH0 for the spread of infection from household to household. The threshold parameters are used to derive the levels of immunity required for the prevention of major epidemics in the community. They are also used to evaluate various vaccination strategies having the same vaccination coverage. For a community with households of equal size, it is found that random vaccination of individuals is better than immunizing all members of a corresponding fraction of households. In contrast, when households have varying sizes, immunizing all members of large households can be better than a corresponding vaccination coverage of randomly selected individuals. It is illustrated that these threshold parameters can also be used for a community of households with schools or day care centers. In particular, the effectiveness of immunizing all members of a school is quantified.

Child↗

PubMed2006

Modelling the relationship between antibody-dependent enhancement and immunological distance with application to dengue.

When antibodies raised in response to a particular pathogen bind with immunologically similar pathogens it may facilitate infection through a phenomenon known as antibody-dependent enhancement (ADE). This process occurs between the four serotypes of dengue virus and, furthermore, secondary infection is a major risk factor in dengue hemorrhagic fever (DHF). Theory has suggested that ADE may be responsible for the large immunological distance between dengue serotypes. We investigate this hypothesis using an epidemic model for dengue in which immunological distance and the strength of immune cross-reaction are expressed separately. Cross-enhancement is considered in three alternative forms acting on susceptibility, transmission and mortality. Previous models have shown that transmission and mortality enhancement can lead to periodicity or chaos. We confirm this result for reasonable levels of susceptibility and transmission enhancement but not for mortality enhancement. We also show that when the two strains have identical basic reproductive numbers no form of enhancement leads to competitive exclusion. When the two strains have different basic reproductive numbers susceptibility or transmission enhancement allow strains with greater immunological similarity to stably coexist but mortality enhancement forces strains to be more distinct. All three forms of enhancement can be associated with DHF and we conclude that mortality enhancement must be dominant if ADE really is responsible for the immunological distance between dengue serotypes.

Antibody-Dependent Enhancement↗

PubMed1996

A model of long-term decline in the transmissibility of an infectious disease: implications for the incidence of hepatitis A.

BACKGROUND: The epidemiology of hepatitis A in countries across the world is changing due to improvements in hygiene and living conditions which reduce the transmissibility of the infection. METHODS: A mathematical model is formulated to describe the changes in incidence of a directly transmitted infection produced by a long term decline in its transmissibility. The basic reproduction number, the parameter describing transmissibility, is considered as a function of time. The relationship between the basic reproduction number and the force of infection is derived. RESULTS: Theoretical examples demonstrate that a decline in transmissibility results in an initial decline in the force of infection, but that this may be followed by a substantial resurgence. Resurgences may be possible after several decades of declining incidence, and are most marked following a rapid decline. CONCLUSIONS: Countries which have experienced a rapid decline in the incidence of hepatitis A may be at risk of a resurgence. More detailed mathematical models, informed with data from regular age-stratified serological surveys, should provide the basis for decisions on vaccination policy.

Age Distribution↗

PubMed2006

How do pathogen evolution and host heterogeneity interact in disease emergence?

Heterogeneity in the parameters governing the spread of infectious diseases is a common feature of real-world epidemics. It has been suggested that for pathogens with basic reproductive number R(0)>1, increasing heterogeneity makes extinction of disease more likely during the early rounds of transmission. The basic reproductive number R(0) of the introduced pathogen may, however, be less than 1 after the introduction, and evolutionary changes are then required for R(0) to increase to above 1 and the pathogen to emerge. In this paper, we consider how host heterogeneity influences the emergence of both non-evolving pathogens and those that must undergo adaptive changes to spread in the host population. In contrast to previous results, we find that heterogeneity does not always make extinction more likely and that if adaptation is required for emergence, the effect of host heterogeneity is relatively small. We discuss the application of these ideas to vaccination strategies.

Adaptation, Biological↗

PubMed1998

Optimal vaccine trial design when estimating vaccine efficacy for susceptibility and infectiousness from multiple populations.

Vaccination can have important indirect effects on the spread of an infectious agent by reducing the level of infectiousness of vaccinees who become infected. To estimate the effect of vaccination on infectiousness, one typically requires data on the contacts between susceptible and infected vaccinated and unvaccinated people. As an alternative, we propose a trial design that involves multiple independent and interchangeable populations. By varying the fraction of susceptible people vaccinated across populations, we obtain an estimate of the reduction infectiousness that depends only on incidence data from the vaccine and control groups of the multiple populations. One can also obtain from these data an estimate of the reduction of susceptibility to infection. We propose a vaccination strategy that is a trade-off between optimal estimation of vaccine efficacy for susceptibility and of vaccine efficacy for infectiousness. We show that the optimal choice depends on the anticipated efficacy of the vaccine as well as the basic reproduction number of the underlying infectious disease process. Smaller vaccination fractions appear desirable when vaccine efficacy is likely high and the basic reproduction number is not large. This strategy avoids the potential for too few infections to occur to estimate vaccine efficacy parameters reliably.

Clinical Trials as Topic↗

PubMed2006

A comparison of continuous and discrete-time West Nile virus models.

The first recorded North American epidemic of West Nile virus was detected in New York state in 1999, and since then the virus has spread and become established in much of North America. Mathematical models for this vector-transmitted disease with cross-infection between mosquitoes and birds have recently been formulated with the aim of predicting disease dynamics and evaluating possible control methods. We consider discrete and continuous time versions of the West Nile virus models proposed by Wonham et al. [Proc. R. Soc. Lond. B 271:501-507, 2004] and by Thomas and Urena [Math. Comput. Modell. 34:771-781, 2001], and evaluate the basic reproduction number as the spectral radius of the next-generation matrix in each case. The assumptions on mosquito-feeding efficiency are crucial for the basic reproduction number calculation. Differing assumptions lead to the conclusion from one model [Wonham, M.J. et al., [Proc. R. Soc. Lond. B] 271:501-507, 2004] that a reduction in bird density would exacerbate the epidemic, while the other model [Thomas, D.M., Urena, B., Math. Comput. Modell. 34:771-781, 2001] predicts the opposite: a reduction in bird density would help control the epidemic.

Algorithms↗

PubMed2002

The general mixing of addicts and needles in a variable-infectivity needle-sharing environment.

In this paper we develop and analyse a model for the spread of HIV/AIDS amongst a population of injecting drug users. The model we discuss focuses on the transmission of HIV through the sharing of contaminated drug injection equipment and in particular we examine the mixing of addicts and needles when the AIDS incubation period is divided into three distinct infectious stages. The impact of this assumption is to greatly increase the complexity of the HIV transmission mechanism. We begin the paper with a brief literature review followed by the derivation of a model which incorporates three classes of infectious addicts and three classes of infectious needles and where a general probability structure is used to represent the interaction of addicts and needles of varying levels of infectivity. We find that if the basic reproductive number is less than or equal to unity then there exists a globally stable disease free equilibrium. The model possesses an endemic equilibrium solution if the basic reproductive number exceeds unity. We then conduct a brief simulation study of our model. We find that the spread of disease is heavily influenced by the way addicts and needles of different levels of infectivity interact.

Computer Simulation↗

PubMed2002

Modelling plant virus epidemics in a plantation-nursery system.

The material used for propagation and planting of many perennial crop plants is derived from vegetative cuttings which are first multiplied in a nursery. This situation was modelled to analyse the dynamics of a plant virus epidemic in a combined nursery-plantation system and the comparative effects of disease management activities in the plantation and in the nursery. The plant populations were partitioned into healthy and diseased categories and were linked according to basic SI models of disease transmission. Removal of diseased plants and replanting operations in both the plantation and nursery were included in the model and two variants were analysed in which mother plants (from which cuttings were taken) remained in the plantation or were harvested. The former is shown to be the limiting case for a large number of cuttings per plant. A criterion was derived for the invasion of disease into a healthy combined system. This consisted of four additive terms: the basic reproductive numbers of disease in the plantation alone and in the nursery alone, and two terms describing the cycling of diseased material between the plantation and nursery. Disease can still invade the system with basic reproductive numbers in the plantation or in the nursery less than 1 depending on the magnitude of cycling. Under some conditions only diseased plants remain in the plantation and nursery. For such a case a criterion was derived for the invasion of healthy plants into a fully diseased system. This depended on replanting rates in the plantation and nursery, and infection, mortality and removal rates of healthy plants.

Disease Outbreaks↗

PubMed2006

Mathematical analysis of a metapopulation model with space-limited recruitment.

We investigate a mathematical aspect of a multi-species' sessile metapopulation model with space-limited recruitment proposed by Iwasa et al. in 1986. We define some basic reproduction numbers to show the threshold condition for the stability of trivial steady state and the existence of coexistent steady state. We show the existence of steady state where all species exist when some reproduction numbers are greater than one by the fixed point theorem. And we construct the Lyapunov function to show the global stability of trivial steady state when some basic reproduction numbers are not greater than one.

Animals↗

PubMed2006

The 1918 influenza A epidemic in the city of São Paulo, Brazil.

The 1918 pandemic H1N1 outbreak in the city of São Paulo is revisited. The outbreak lasted for 10 weeks and reached 116,771 officially recorded cases amongst 523,194 inhabitants. The total number of deaths summed up to 5331, with a lethality rate of 4.5% and an overall mortality rate of around 1%. We propose a mathematical model that tallies available data with good accuracy and allows the estimation of the basic reproductive number, R(0). The model showed a remarkably good accuracy in retrieving the real data from São Paulo city outbreak considering the total number of recorded cases and deaths and the timing of the outbreak. The basic reproduction number calculated of 2.68 can be compared to estimates carried out for other flu strains, like the estimates for H3N2, whose values ranged from 1.5 to 2.5. We hypothesize that the Southern parts of the world in which there was relatively little impact of the Great War, like South America, suffered a much lower H1N1 influenza mortality as compared with that reported for the Northern hemisphere heavily affected by the I World War.

Brazil↗

PubMed2004

Modelling the social dynamics of a sex industry: its implications for spread of HIV/AIDS.

A theoretical model is proposed for a community which has the structure of two classes (direct and indirect) of commercial sex workers (CSW), and two classes of sexually active male customers with different levels of sexual activity. The direct CSW's work in brothels while the indirect CSW's are based in commercial establishments such as bars, cafes, and massage parlours where sex can be bought on request and conducted elsewhere. Behaviour change and the resulting change of activity class occurs between the two classes of CSW's and two classes of males under the setting of the proliferation of human immunodeficiency virus (HIV)/acquired immunodeficiency syndrome epidemic and the subsequent intervention programmes. In recently years, this phenomenon has been observed in several countries in Asia. Given the lower levels of condom use and higher HIV prevalence of the indirect CSW's, ascertaining the impact of this change in the structure of the sex industry on the spread of HIV is the main focus of this paper. The complete analysis of the disease-free model is given. For the full model, local analysis will be performed for the case of preferred mixing without activity class change, as well as the case with activity class change and restricted mixing. The basic reproduction number for the spread of epidemic will be derived for each case. Results of biological significance include: (i). the change of behaviour by the CSW's has a more direct effect on the spread of HIV than that of the male customers; (ii). the basic reproduction number is obtained by considering all possible infection cycles of the heterosexual transmission of HIV which indicates the importance of understanding the sexual networking in heterosexual transmission of HIV; (iii). the social dynamics of the sex industry is not just a simple 'supply and demand' mechanism driven by the demand of the customers, hence highlighting the need for further understanding of the changing structure of the sex industry. The main points of this work will be illustrated with numerical examples using the HIV data of Thailand.

Disease Transmission, Infectious↗

PubMed1999

Vancomycin-resistant enterococci in intensive-care hospital settings: transmission dynamics, persistence, and the impact of infection control programs.

Vancomycin-resistant enterococci (VRE) recently have emerged as a nosocomial pathogen especially in intensive-care units (ICUs) worldwide. Transmission via the hands of health-care workers is an important determinant of spread and persistence in a VRE-endemic ICU. We describe the transmission of nosocomial pathogens by using a micro-epidemiological framework based on the transmission dynamics of vector-borne diseases. By using the concept of a basic reproductive number, R0, defined as the average number of secondary cases generated by one primary case, we show quantitatively how infection control measures such as hand washing, cohorting, and antibiotic restriction affect nosocomial cross-transmission. By using detailed molecular epidemiological surveillance and compliance monitoring, we found that the estimated basic reproductive number for VRE during a study at the Cook County Hospital, Chicago, was approximately 3-4 without infection control and 0.7 when infection control measures were included. The impact of infection control was to reduce the prevalence from a predicted 79% to an observed 36%. Hand washing and staff cohorting are the most powerful control measures although their efficacy depends on the magnitude of R0. Under the circumstances tested, endemicity of VRE was stabilized despite infection control measures, by the constant introduction of colonized patients. Multiple stochastic simulations of the model revealed excellent agreement with observed pattern. In conjunction with detailed microbiological surveillance, a mathematical framework provides a precise template to describe the colonization dynamics of VRE in ICUs and impact of infection control measures. Our analyses suggest that compliance for hand washing significantly in excess of reported levels, or the cohorting of nursing staff, are needed to prevent nosocomial transmission of VRE in endemic settings.

Anti-Bacterial Agents↗

PubMed1999

Chlamydia transmission: concurrency, reproduction number, and the epidemic trajectory.

To identify factors that influence individual and group transmission of Chlamydia, the authors conducted community-wide contact tracing of chlamydia cases in Colorado Springs, Colorado, from mid-1996 to mid-1997. Case patients identified persons with whom they had had contact during the 6 months preceding diagnosis; contacts were actively sought and offered DNA amplification testing. Sexual contact networks were used to identify "source cases" and "spread cases," permitting estimation of the basic reproduction number (R0) for individuals and groups. Network and epidemiologic factors influencing R0 were assessed using univariate and multivariate procedures. Of 1,309 case patients, 1,131 (86%) were interviewed, and 2,409 contacts were identified. The 1,131 interviewed cases yielded 623.9 computed spread cases, for an overall R0 of 0.55. Few subgroups analyzed yielded a mean R0 exceeding unity-an observation in keeping with routine surveillance information which suggests that chlamydia incidence is declining in Colorado Springs. Concurrency, a network measure of simultaneous partnerships, was the most powerful predictor of transmission. Direct estimation of basic reproduction numbers for chlamydia using contact tracing techniques is feasible and can produce useful data with which to prioritize control efforts, evaluate interventions, and gauge the place of chlamydia on the epidemic continuum.

Adolescent↗

PubMed2002

Model-based estimation of vaccine effects from community vaccine trials.

Community vaccine trials are becoming increasingly important to assess both the direct and indirect community level effects of vaccination. In this paper, we present statistical methods to analyse such trials, using a design with several matched pairs of communities. The communities are matched on similarities in infection transmission as reflected through the basic reproduction number. Two methods of analysis are presented and compared. The first is simple empirical estimation of vaccine effects. Summary measures of these effects are constructed by reciprocal variance weighted averages across the community pairs. The second is likelihood-based where we derive a mixed effects epidemic model. This model takes the intercommunity variability into account through a random effect on the basic reproduction number. With this model, we derive a distribution-free estimator for the variance of the random effect. We use simulated epidemics to explore the performance of the two estimation methods for different numbers of community pairs and different levels of inter-pair variability. Both methods provide acceptable estimates in terms of bias and precision under reasonable conditions. Although the empirical approach involves fewer assumptions than the model-based approach, the resulting vaccine effectiveness estimates are only applicable to the vaccination fraction tested in the trial. In contrast, the model-based approach can be used to predict the vaccine effectiveness at vaccination fractions other than those used in the trial. Thus, it can be used as a public health policy tool for predicting the community level effects of vaccination. We demonstrate such use by predicting total vaccine effectiveness for the whole range of vaccination fractions.

Computer Simulation↗

PubMed2003

Global dynamics of an SEIR epidemic model with saturating contact rate.

Heesterbeek and Metz [J. Math. Biol. 31 (1993) 529] derived an expression for the saturating contact rate of individual contacts in an epidemiological model. In this paper, the SEIR model with this saturating contact rate is studied. The basic reproduction number R0 is proved to be a sharp threshold which completely determines the global dynamics and the outcome of the disease. If R0 < or =1, the disease-free equilibrium is globally stable and the disease always dies out. If R0 > 1, there exists a unique endemic equilibrium which is globally stable and the disease persists at an endemic equilibrium state if it initially exists. The contribution of the saturating contact rate to the basic reproduction number and the level of the endemic equilibrium is also analyzed.

Disease Outbreaks↗

PubMed1999

Transmission dynamics and epidemiology of dengue: insights from age-stratified sero-prevalence surveys.

The relationship between infection with the four major serotypes of dengue virus and the occurrence of different forms of disease is complex and not fully understood. Interpreting longitudinal records of the incidence of serious disease to gain insight into the transmission dynamics and epidemiology of the virus is therefore complicated. Since age reflects duration of exposure, age-stratified, strain-specific serological surveys carried out at one point in time, or over a short time interval, can potentially provide a rich source of information on longitudinal patterns. This paper describes the development and application (to data collected in Thailand) of statistically rigorous methods designed to estimate time-varying, strain-specific forces of infection, and thus basic reproduction numbers, from cross-sectional serological data. The analyses provide support for the hypothesis that antibody-dependent enhancement of transmission influences observed epidemiological pattern. Age-stratified serological data also reveal evidence of a propensity for the annual incidence of infection to oscillate over time with a frequency of several years. The latter observation is consistent with the predictions of simple mathematical models of the transmission dynamics of the virus. The estimates of the basic reproduction numbers obtained are similar in magnitude for each dengue serotype, being in the range of four to six. Such values are higher than those obtained from earlier analyses, and the implications of this for dengue control are discussed.

Dengue↗